Question

(a) (b) (c) (d) (e) Let A, B, and C be sets. Use methods of Example 10.25 to prove the following set equations. An(B \ C) = (A n B) \ C (A \ C) n B. Au(B \ C) = (A U B) \ (CIA). (AUB)\C-(A\C)U(B\C) (A B) \ C = A n (B \ C) = (A \ C) n B. (4)B) \ C = A \ (BUC).

Answer 1

The student's question involves proving identities in set theory using set operations like intersection (n), union (u), and difference (\). By logically analyzing the relationships between sets and applying set operators, one proves that elements of one set expression are equal to another set expression.

Step-by-step explanation:

The question you've asked relates to set theory, and it involves proving identities involving set operations such as union, intersection, and difference. Let's consider the first equation you've provided:

An(B \ C) = (A n B) \ C

To prove this, we will show that each element of An(B \ C) is also in (A n B) \ C and vice versa.

Let x be an element of An(B \ C). This means that:

• x is in A.
• x is in B, but not in C.

Since x is in A and B, x is also in A n B. However, because x is not in C, x is also in (A n B) \ C, which proves that An(B \ C) is a subset of (A n B) \ C. Similarly, we can take any element from (A n B) \ C and show that it must be in An(B \ C), proving the two sets are equal and the identity holds.

This approach can be applied to the rest of the set equations by analyzing the relationships between the sets and applying logical reasoning corresponding to each set operator.